3.985 \(\int (1-x)^{-\frac{1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac{1}{2}+p} \, dx\)

Optimal. Leaf size=83 \[ -\frac{4^{p+1} (1-x)^{p+\frac{1}{2}} \left (\frac{x}{x+1}\right )^{2 (p+1)} (x+1)^{p+\frac{3}{2}} (c x)^{-2 (p+1)} \, _2F_1\left (p+\frac{1}{2},2 (p+1);p+\frac{3}{2};\frac{1-x}{x+1}\right )}{2 p+1} \]

[Out]

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*(1 + p))*(1 + x)^(3/2 + p)*Hyperge
ometric2F1[1/2 + p, 2*(1 + p), 3/2 + p, (1 - x)/(1 + x)])/((1 + 2*p)*(c*x)^(2*(1
+ p))))

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Rubi [A]  time = 0.0755755, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ -\frac{4^{p+1} (1-x)^{p+\frac{1}{2}} \left (\frac{x}{x+1}\right )^{2 (p+1)} (x+1)^{p+\frac{3}{2}} (c x)^{-2 (p+1)} \, _2F_1\left (p+\frac{1}{2},2 (p+1);p+\frac{3}{2};\frac{1-x}{x+1}\right )}{2 p+1} \]

Antiderivative was successfully verified.

[In]  Int[((1 - x)^(-1/2 + p)*(1 + x)^(1/2 + p))/(c*x)^(2*(1 + p)),x]

[Out]

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*(1 + p))*(1 + x)^(3/2 + p)*Hyperge
ometric2F1[1/2 + p, 2*(1 + p), 3/2 + p, (1 - x)/(1 + x)])/((1 + 2*p)*(c*x)^(2*(1
+ p))))

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Rubi in Sympy [A]  time = 5.89622, size = 71, normalized size = 0.86 \[ - \frac{\left (c x\right )^{- 2 p - 1} \left (\frac{x + 1}{- x + 1}\right )^{- p - \frac{1}{2}} \left (- x + 1\right )^{p + \frac{1}{2}} \left (x + 1\right )^{p + \frac{1}{2}}{{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p - \frac{1}{2} \\ - 2 p \end{matrix}\middle |{- \frac{2 x}{- x + 1}} \right )}}{c \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(-1/2+p)*(1+x)**(1/2+p)/((c*x)**(2+2*p)),x)

[Out]

-(c*x)**(-2*p - 1)*((x + 1)/(-x + 1))**(-p - 1/2)*(-x + 1)**(p + 1/2)*(x + 1)**(
p + 1/2)*hyper((-2*p - 1, -p - 1/2), (-2*p,), -2*x/(-x + 1))/(c*(2*p + 1))

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Mathematica [A]  time = 0.211395, size = 75, normalized size = 0.9 \[ -\frac{\left (\frac{1-x}{x+1}\right )^{-p-\frac{1}{2}} \left (1-x^2\right )^{p+\frac{1}{2}} (c x)^{-2 p-1} \, _2F_1\left (-2 p-1,\frac{1}{2}-p;-2 p;\frac{2 x}{x+1}\right )}{2 c p+c} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((1 - x)^(-1/2 + p)*(1 + x)^(1/2 + p))/(c*x)^(2*(1 + p)),x]

[Out]

-(((c*x)^(-1 - 2*p)*((1 - x)/(1 + x))^(-1/2 - p)*(1 - x^2)^(1/2 + p)*Hypergeomet
ric2F1[-1 - 2*p, 1/2 - p, -2*p, (2*x)/(1 + x)])/(c + 2*c*p))

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Maple [F]  time = 0.201, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( cx \right ) ^{2+2\,p}} \left ( 1-x \right ) ^{-{\frac{1}{2}}+p} \left ( 1+x \right ) ^{{\frac{1}{2}}+p}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x)

[Out]

int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x\right )^{-2 \, p - 2}{\left (x + 1\right )}^{p + \frac{1}{2}}{\left (-x + 1\right )}^{p - \frac{1}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2),x, algorithm="maxima")

[Out]

integrate((c*x)^(-2*p - 2)*(x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (x + 1\right )}^{p + \frac{1}{2}}{\left (-x + 1\right )}^{p - \frac{1}{2}}}{\left (c x\right )^{2 \, p + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2),x, algorithm="fricas")

[Out]

integral((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**(-1/2+p)*(1+x)**(1/2+p)/((c*x)**(2+2*p)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{p + \frac{1}{2}}{\left (-x + 1\right )}^{p - \frac{1}{2}}}{\left (c x\right )^{2 \, p + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2),x, algorithm="giac")

[Out]

integrate((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2), x)